BX000A

binary_modular_square_residue_exists

Every squaring step has an actual canonical modular residue.

Alpha v34 checked-use · first admitted v21 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

G102 was OPEN when this family was first admitted in Alpha v21. It is now CLOSED in Alpha v23: every arbitrary exponent has actual canonical beta-coded digits, a complete modular execution, and exact counted bound operations≤3*BitLen(e)+2.

Exact theorem in conservative defined notation

∀ m. ∀ x. BinaryModulus(m) → ∃ y. CanonicalModularResidue(m,x · x,y)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

BinaryModulus(m) · 1CanonicalModularResidue(m,a,r) · 1

Actual proof prerequisites

binary_canonical_residue_exists
Original expanded first-order statement 
forall m x. (exists ff_modulus_gap_binary_guard. ff_modulus_gap_binary_guard + S 1 = m) -> exists r. (((exists ff_gap_binary_squared. ff_gap_binary_squared + S (r) = m) /\ (exists ff_left_binary_squared_congruence ff_right_binary_squared_congruence. (x * x) + m * ff_left_binary_squared_congruence = (r) + m * ff_right_binary_squared_congruence)))

Complete unchanged native tactic proof

All 7 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

7 script commands · 2 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (1)
binary canonical residue exists
01Fix variables and assumptionsL1–3

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro m
  2. L2
    intro x
  3. L3
    intro hmodulus
02Use earlier factsL4–7

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L4
    specialize binary_canonical_residue_exists m
  2. L5
    specialize binary_canonical_residue_exists (x * x)
  3. L6
    apply binary_canonical_residue_exists
  4. L7
    exact hmodulus

Library-wide reading audit

Original defined command ledger · 7 lines
  1. 0001intro m
  2. 0002intro x
  3. 0003intro hmodulus
  4. 0004specialize binary_canonical_residue_exists m
  5. 0005specialize binary_canonical_residue_exists (x * x)
  6. 0006apply binary_canonical_residue_exists
  7. 0007exact hmodulus